Two kings battling and both are extremely renowned sword fighters throughout their stories. Both from their books; Arthur from the original myths and Aragorn from the LOTR books. Arthur with Excalibur, the sword in the stone, and Aragorn with Anduril, flame of the west. Neither get help of any sort. The battlefield is an open plain with no cover. Both have only their swords as weapons; no bows, no dagger, etc.

Aragorn the character appears to be indebted to the legendary Arthur. Both were Once and Future Kings of prophecy and birthright. Both had special swords as mentioned in the OP. Both were renowned kings, champions, heroes, and leaders of men. For the sake of this match up, let us assume their skill with the sword was comparable and that their respective swords were equal in potency.

As written and as portrayed in the adaptations, Aragorn is better in every way.

Let's start with this comparison: What did each one have to do to win his crown? Starting with Aragorn.

Aragorn had to go through some arduous tasks that required much valor. He helped Frodo deliver the Ring to Mordor, or at least to Parth Galen at any rate, and eventually lead the Men of the West to distract the Enemy to allow Frodo to finish the job. Through his leadership, he ensured that there would still be Men of the West, and by extension, all Free Peoples, in existence for long enough to allow that quest to be fulfilled. He fought and led at Helms Deep and successfully protected Rohan from extinction. He led the Grey Company down the Paths of the Dead and rallied the Oathbreakers and the forces of Gondor. He commandeered the Corsair vessels and won the Pelennor Fields. He brought the War to Sauron's doorstep and saw the end of the Shadow. Only after all that did he earn his crown.

What did Arthur do to win his crown? he pulled a sword out of a freakin' stone.

Aragorn had better moral character, for another thing, and therefore more likely to have greater morale and righteous determination to give him an edge in the duel. Let's not forget what Arthur did to all those May babies (unsuccessfully trying to prevent the prophecy relating to Mordred). I'm not even going to bring up the versions of the legend where he commits rape and incest (except that I just mentioned it. oops.)

Also, Aragorn enjoys the advantage of some Elvish ancestry, as with all Numenoreans, giving him special longevity and long life. What does Arthur have? Possibly a connection to Avalon (Faerie), depending on the version of the legend, but its a bit vague. Someone else mentioned correctly that Aragorn, in his prime, was older and had more life experience , and battle experience, than Arthur in his prime, and so that favors Aragorn in this matchup.

Aragorn is probably more intelligent. He speaks fluent Elvish, is skilled as an outdoorsman, tracker, hunter, and herbalist, and knows the arts of healing and treatment against Morgul sorcery. Arthur may have been fluent in other languages, like maybe French, but I'm not aware of any other special knowledge that he was famous for having or using.

Aragorn was a skilled equestrian. Arthur was probably ok as well, but Aragorn was particularly good at it, especially as shown in TTT:EE. Did Arthur even have a particular horse? I don't recall any. Win goes to Aragorn here.

It's not even close. Aragorn!!! A bag is like a hole that you can carry with you.

Are the two swords essentially equals? Or is one superior to the other? Do they make a difference in the confrontation of the two kings? 'There are older and fouler things than Orcs in the deep places of the world.' - Gandalf the Grey, The Fellowship of the Ring

Both are strong, well-forged weapons with magical properties and mystical heritage. Anduril was forged by the Elves and powerful enough to cut the Ring from Sauron's hand. Excalibur was forged by the people of Avalon and was said to flash so brightly, it could blind Arthur's enemies.

There are two aspects that could determine which sword is better. First, Excalibur's scabbard would give Arthur an advantage. It was enchanted and could prevent blood loss from battle wounds. But it was later stolen by Morgan le Fay and thrown into a lake. Does Arthur have it in this battle? Or do he and Aragorn just have their swords?

Second, the "pedigree" of the magic would depend on which version of Avalon we're using. In some, it's synonymous with the Faerie Realm. The Lady of the Lake, who bestows Excalibur, is sometimes considered to be a faerie. This would put her magic on par with the Elves. In other versions though, the people of Avalon are mortal, simply a druidic order or a group of sorceresses. Is human magic as strong as that of the Elves?

This ultimately is the issue in determining an Arthur vs. Aragorn battle. There isn't a single text for the Arthur legend that can serve as the absolute authority. There are many different versions.